Travelling-wave Perturbations Research Articles

Roughness on liquid-infused surfaces induced by capillary waves

Abstract

Front transition in higher order diffusion equations with a general reaction nonlinearity

In this paper, we investigate the wave front solutions of a class of higher order reaction diffusion equations with a general reaction nonlinearity. Linear stability analysis with a modulated travelling wave perturbation is used to prove the existence of wave front solutions. We proved that the studied equation supports both monotonic translating front and patterned front solutions. Also, a minimal front speed and the condition for a transition between these front types (monotonic and patterned) are determined. Two numerical examples are discussed (the extended Fisher-Kolmogorov equation with two different reaction nonlinearities) to support the obtained results.

Linear instability of low Reynolds number massively separated flow around three NACA airfoils

Two- and three-dimensional modal and non-modal instability mechanisms of steady spanwise-homogeneous laminar separated flow over airfoil profiles, placed at large angles of attack against the oncoming flow, have been investigated using global linear stability theory. Three NACA profiles of distinct thickness and camber were considered in order to assess geometry effects on the laminar–turbulent transition paths discussed. At the conditions investigated, large-scale steady separation occurs, such that Tollmien–Schlichting and cross-flow mechanisms have not been considered. It has been found that the leading modal instability on all three airfoils is that associated with the Kelvin–Helmholtz mechanism, taking the form of the eigenmodes known from analysis of generic bluff bodies. The three-dimensional stationary eigenmode of the two-dimensional laminar separation bubble, associated in earlier analyses with the formation on the airfoil surface of large-scale separation patterns akin to stall cells, is shown to be more strongly damped than the Kelvin–Helmholtz mode at all conditions examined. Non-modal instability analysis reveals the potential of the flows considered to sustain transient growth which becomes stronger with increasing angle of attack and Reynolds number. Optimal initial conditions have been computed and found to be analogous to those on a cascade of low pressure turbine blades. By changing the time horizon of the analysis, these linear optimal initial conditions have been found to evolve into the Kelvin–Helmholtz mode. The time-periodic base flows ensuing linear amplification of the Kelvin–Helmholtz mode have been analysed via temporal Floquet theory. Two amplified modes have been discovered, having characteristic spanwise wavelengths of approximately 0.6 and 2 chord lengths, respectively. Unlike secondary instabilities on the circular cylinder, three-dimensional short-wavelength perturbations are the first to become linearly unstable on all airfoils. Long-wavelength perturbations are quasi-periodic, standing or travelling-wave perturbations that also become unstable as the Reynolds number is further increased. The dominant short-wavelength instability gives rise to spanwise periodic wall-shear patterns, akin to the separation cells encountered on airfoils at low angles of attack and the stall cells found in flight at conditions close to stall. Thickness and camber have quantitative but not qualitative effect on the secondary instability analysis results obtained.

Adjustment of spiral drift by a travelling wave perturbation

We apply a velocity-field approach to investigate the interaction between spiral waves and the travelling wave modulation of system excitability which leads to a prediction: the direction of the straight-line drift of spiral waves is linearly adjusted by the propagation direction of the travelling waves. Direct numerical computations of the Oregonator model and the formulas of drift-velocity field confirm the validity and robustness of our theoretical prediction.

Dynamics of Vortex-Wave under a Travelling-Wave Modulation

The mechanism of destabilization is studied for the rotating vortices (scroll waves and spiral waves) in excitable media induced by a parameter modulation in the form of a travelling-wave. It is found that a rigid rotating spiral in the two-dimensional (2D) system undergoes a synchronized drift along a straight line, and a 3D scroll ring with its filament closed into a circle can be reoriented only if the direction of wave number of a travelling-wave perturbation is parallel to the ring plane. Then, in order to describe the behaviour of the synchronized drift of spiral wave and the reorientation of scroll ring, the approximate formulas are given to exhibit qualitative agreements with the observed results.

Control of spiral waves and turbulent states in a cardiac model by travelling-wave perturbations

We propose a travelling-wave perturbation method to control the spatiotemporal dynamics in a cardiac model. It is numerically demonstrated that the method can successfully suppress the wave instability (alternans in action potential duration) in the one-dimensional case and convert spiral waves and turbulent states to the normal travelling wave states in the two-dimensional case. An experimental scheme is suggested which may provide a new design for a cardiac defibrillator.

Chaotic Hamiltonian dynamics of particle's horizontal motion in the atmosphere

A non-separable, near-integrable, two degrees-of-freedom (d.o.f.) Hamiltonian system arising in the context of particle's dynamics on a geopotential of the atmosphere is studied. In the unperturbed system, homoclinic orbits to periodic motion in the longitude angle exist, giving rise to several types of homoclinic chaos in the perturbed system; a new type of homoclinic chaos is found, exhibiting chaos which is non-uniformly distributed in the longitude angle along the homoclinic loop while it is topologically uniformly distributed in this angle in a neighborhood of the hyperbolic periodic orbit, i.e. the return map to different values of the longitude angle are topologically conjugate only near the hyperbolic periodic orbit. The concept of colored energy surfaces and its corresponding energy-momentum map are developed as analytical tools for delineating the regions in the four-dimensional phase space where the various types of homoclinic chaos prevail. Geometrical interpretation of the Melnikov analysis is offered for detecting this non-uniformity in angle. Applying the results of the analysis to the planetary atmosphere with an infinite wavelength perturbation of amplitude ϵ, we find that for eastward going particles with initial velocities {(ifu 0, v 0) = (ū, 0)} the chaotic band occupies two narrow (O(ϵ)) bands on both sides of the equator nea latitudes φ = ± arccos(1 − 2 u) . By contrast, for westward going particles the chaotic zone is thicker ( O( ϵ; ) ) and is centered on the equator. Moreover, the dependence of the chaotic zone's thickness on the perturbation frequency is much more sensitive to the value of the initial speed for eastward going particles than for those going westward. Qualitative and quantitative differences between the distribution of the homoclinic chaotic motion in the longitude angle for westward and eastward travelling wave perturbations are predicted and numerically confirmed.

EXPERIMENTAL INVESTIGATION OF STABILITY OF PLANAR CRYSTAL-MELT INTERFACE AND EVOLUTION OF CELLULAR INTERFACE DURING CZOCHRALSKI GROWTH OF LiNbO3 SINGLE CRYSTALS DOPED WITH YTTRIUM

By using periodic rotational striations as a time marker, the initial instability of a planar interface and the development of the cellular interface have been studied in Czochralski growth system of anisotropic LiNbO3 crystals doped with yttrium. The critical condition for very birth of instability of planar interface has been obtained. And there are two kinds of initial perturbations to be observed, that is, sinusoidal perturbation and sinusoidal travelling wave perturbation. Experimental results revealed that the planar interface evolves from sinusoidal perturbation to facetted perturbation to coarse facetted perturbation, finally, to stable cellular interface. It has been found that the wave-length of stable cellular interface is different from that of initial perturbation and always integral times larger than that of initial one. There is also a difference to be found between critical velocity of the plane-to-cell transition and that of cell-to-plane transition for a same crystal grown from the same system. It may be implied that the facetted cell is more stable than the nonfacetted plane.

Gravitational and sound waves in stiff matter

The dynamics of a perfect fluid with pressure equal to energy density is considered in space-times with planar, pseudoplanar, cylindrical and toroidal symmetries. A scheme for obtaining the general exact solution of the field equations is described. It is shown that this scheme admits inclusion of additional massless, minimally coupled scalar fields and (for planar symmetry) electromagnetic plane waves. Two physical examples are discussed: finite perturbations of a static regular cylinder, and waves of finite magnitude in a Kasner-like universe. In the first case it is shown that all physically allowed perturbations preserving cylindrical symmetry are of standing-wave type and cannot propagate. Thus a perturbed static stiff cylinder cannot radiate the Einstein-Rosen waves. In the second case it is shown that travelling-wave perturbations obey a sort of momentum-conservation law; besides, it is argued that sound perturbations can provide a mechanism for fragmentation in a universe filled with stiff matter.

Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation

The stability of plane, periodic solutions of the two-dimensional nonlinear Schrödinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik [8], Saffman and Yuen [4] and Ablowitz and Segur [1] on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities]may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations.

Characteristics of travelling ionospheric perturbations over Waltair using phase path technique

The phase path technique is a very sensitive experimental method for detecting travelling ionospheric perturbations in the FZ-region and is capable of giving much information on the various characteristics of the perturbations. The travelling wave perturbations over Waltair, scaled from phase path records of ‘O’ and ‘X’ components in the F2-region echoes on 5.6 MHz were found to have periods in the range 3–39 min. Smaller periods were of more frequent occurrence than larger periods and the most frequently observed period was about 9 min. An average wave perturbation in the F2-region could be represented by the equation A = 46.49 K sin 2 πt/ K where ‘A’ is in meters, K the period of the disturbances in the general range 3–39 min and ‘ t’ is the time in minutes. The phase speed of the perturbations was in the range 2.5–14 km/min. Perturbations with shorter periods were found to travel faster than larger periods. It was shown that the above variation of the speed of the perturbations with period leads to the conclusion that the group speed of the perturbations is twice their phase speed.

XXXV. Comments on Magnetron Theory, with Particular Reference to Some Recent Publications

ABSTRACT The wartime development of magnetron theory is outlined, beginning with non-existence proofs for double stream steady states, the vindication of the single stream steady state by transient calculations and the recognition that cylindrical symmetry must be abandoned for further progress. The success of work with transverse fields is recalled, giving agreement with experiments in the form of acceptable self-consistent orbits and fields. the threshold criterion and the natural instability of the single stream steady state to travelling wave perturbations, explaining start-up, high noise and high diode currents after ‘cut-off’. Recent publications, all unduly preoccupied with symmetrical steady states, are then discussed, in particular the first three papers of this Journal's first issue. Their writers, perhaps unaware of some of the wartime work because of its inaccessibility, draw conclusions which are at variance with the picture developed during the war. However, closer examination shows that the...